Wednesday, June 17, 2015

What do you mean "finite", kemosabe?

There are a lot of alternatives to what mathematics is. Most of these approaches privilege certain sections of classical mathematics over others, the privileged sections are called "true" and the unprivileged "unproven". Some of them, like the varieties of constructivism or intuitionism, are respectable minority viewpoints. They have disadvantages - the theories aren't closed under as many operations. But they have advantages too, as a constructive proof allows one to actually calculate an answer. This is of great interest for an applied mathematician like myself. Then there are the finitists.

Ever heard of Archimedes, Newton, Cantor? Morons.

Well, it's difficult to talk about finitists, because the bold, bold declarations (Edward Nelson is gonna prove Peano inconsistent! Doubt it) and monster raving egomania (has to be read to be believed) make it hard to understand them. But let's set aside all the technical machinery that masks their real arguments. Let's focus on something so simple even finitist will have trouble corrupting it.

This version is much better than the Three Dog Night one

Is 1 a finite number? What makes you say so? Isn't it a real? Isn't it a limit of the sequence [.9, .99, .999, ...]? Isn't it an infinite amount of zeros, a 1 and decimal marker then another infinite sequence of zeros? Why do we privilege "-1" as a symbol for the number 2 less than one, and not the p-adic representation ...1111111111111? There is no answer. Indeed, in computers that run 2's complement arithmetic, the p-adic representation would be the natural extension, not the invention of this wholly new symbol "-". Whether an object is "finite" or "infinite" is dependent on representation. This is not good!

Right Isosceles Triangle

Draw a line, then another line of the same length at a right angle to that line (this is easy to do if you know a bit about geometry). An amazing thing is true. You can double, triple, quadruple the length of AC easily with just a straight-edge and a ruler. The same is true of BC. But, no matter how much you do so, the two endpoints will never both lie on a circle centered at C. Ever. Try it, it's fun! This means that the side is irrational, and in one representation - decimal - "infinite". As infinite as 1.00000000..! None of the prevarications of Zeilberg or any finitist will get a around that.

"What, my representations arbitrary?"

Look, one could try to argue until one is blue in the face, but finitists don't care about such things. They want there to be a distinction between the finite and the infinite, and they want to stay on this side of it. The only answer the preserves the intuition is that there must be a non-arbitrary representation. And perhaps there is - the representation in a Turing Machine for instance. What's representable in one ought be in another, na? Never you mind that nobody in the world thought like this before and nothing bad has ever come of it, that's pointless twibbling. The issue here is that one does not get finitism out of this line of thought, but rather a form of constructivism. Any alternative to mainstream math thinking is going to lead out of finitism eventually, because finitism is too arbitrary. I've discussed how Wittgenstein's ideas lead to a non-finitist criticism of mainstream bath before.

I missed yesterday, so I'm gonna try to get two posts in today. This blog is officially back in business!